Inflation Slow-Roll Parameter Calculator

Slow-Roll Dynamics in Inflationary Cosmology

The inflationary paradigm proposes that the early universe underwent a brief epoch of accelerated expansion driven by the potential energy of a scalar field, often called the inflaton. During this phase the scale factor grows almost exponentially, smoothing away pre-existing curvature and helping explain why the observable universe looks so homogeneous on large scales. At the same time, tiny quantum fluctuations are stretched to astrophysical sizes and later become the seeds of galaxies and cosmic structure. The slow-roll approximation is the standard first tool for studying this process because it turns the field equations into simple relations between the shape of the potential and observable quantities.

Introduction

This calculator focuses on monomial inflationary potentials of the form $V\left(\phi \right)=\lambda {\phi }^n$. In plain language, that means the potential energy is proportional to a power of the field value. By choosing the power n, a field value $\phi$, and a number of e-folds $N$, you can estimate the first slow-roll parameter $\epsilon$, the second slow-roll parameter $\eta$, the scalar spectral index $n_s$, and the tensor-to-scalar ratio $r$. These are the quantities most often compared with cosmic microwave background observations.

In canonical single-field inflation the first slow-roll parameter is defined as $\epsilon =\frac{M^2}{2}\left(\frac{V\text{'}}{V}\right)$. It measures how steep the potential is relative to its height. Inflation requires $\epsilon \ll 1$; once $\epsilon$ approaches unity, accelerated expansion ends. The second slow-roll parameter $\eta$ tracks the curvature of the potential and helps determine how the primordial power spectrum tilts away from exact scale invariance.

Observables such as the scalar spectral index $n_s\approx 1-6\epsilon +2\eta$ and the tensor-to-scalar ratio $r\approx 16\epsilon$ are then obtained at leading order. Because modern observations constrain both $n_s$ and $r$, even a simple calculator like this one is useful for building intuition about which inflationary models remain plausible and which are under pressure from data.

How to Use

Use the form below as a quick exploratory tool. The calculator preserves the original interactive behavior of the page: it reads the values you enter for the potential power, field value, and e-fold count, then computes the corresponding slow-roll quantities directly from the formulas in the script. Although the page also mentions a back-solved field estimate from $N$, the live result shown in the output box is based on the field value $\phi$ that you type into the form.

Here is what each input means in practice. The potential power n sets the shape of the monomial potential. A larger value of n makes the potential steeper for large field values and usually increases the predicted tensor signal. The field value φ is entered in reduced Planck mass units, so it is dimensionless in the conventions used by the calculator. The e-folds N input is included because it is a standard inflationary parameter and is used by the table examples and helper function in the script, even though the main button calculation uses your typed $\phi$ directly.

To get a result, enter numeric values and press Compute Parameters. The output area will report $\epsilon$, $\eta$, $n_s$, and $r$, followed by a short interpretation of whether the tensor ratio is below the page's stated comparison threshold of . If you enter an invalid value, such as a non-numeric entry or a non-positive field value, the calculator will show an error message instead of a result.

For best use, treat this page as a teaching and estimation tool. Try changing one input at a time. If you increase $\phi$ while keeping n fixed, both $\epsilon$ and $\eta$ usually decrease because the denominators contain ${\phi }^2$. If you increase n at fixed $\phi$, the potential becomes effectively steeper and the predicted tensor ratio tends to rise. Watching those trends is often more informative than any single numerical output.

Formula

For a monomial potential, the derivatives are simple enough that the slow-roll expressions can be written in closed form. The page script computes the main output from the field value you provide using

Formula: ε = n^2 / (2 φ^2)

$\epsilon =\frac{n^2}{2{\phi }^2}$

Formula: η = (n(n − 1)) / φ^2

$\eta =\frac{n(n-1)}{{\phi }^2}$

Formula: n_s = 1 − 6 ε + 2 η

$n_s=1-6\epsilon +2\eta$

Formula: r = 16 ε

$r=16\epsilon$

The page also includes a helper relation for estimating the field value associated with a chosen number of e-folds in a monomial model. In the slow-roll approximation, the number of e-folds between a field value and the end of inflation is approximately $N\approx \frac{{\phi }^2-{\phi }_{\mathrm{end}}^2}{2n}$. The script uses the approximation $\phi \approx \sqrt{2nN+\frac{n^2}{2}}$ when filling the example table. That means the table and the button output are related but not identical: the table derives $\phi$ from $N$, while the main calculator uses the field value you enter manually.

This distinction matters for interpretation. If you want the result to correspond to a specific e-fold count in the same approximation used by the table, choose a field value consistent with that relation. If instead you are exploring a field point directly, the calculator still gives the correct leading-order slow-roll quantities for that chosen $\phi$ and n.

Worked Example

Suppose you choose a quadratic potential with n = 2 and enter a field value $\phi =5$ in reduced Planck units. The first slow-roll parameter becomes $\epsilon =\frac{4}{2\times 25}=0.08$. The second parameter is $\eta =\frac{2}{\times }$, which is $0.08$ as well. Then the scalar spectral index is $n_s=1-6\times 0.08+2\times 0.08=0.68$, and the tensor-to-scalar ratio is $r=16\times 0.08=1.28$.

That example is intentionally simple, and it also shows why interpretation matters. A result like $r=1.28$ is far above current observational bounds, so the page will flag it as inconsistent with the comparison threshold. The lesson is not that quadratic inflation is always impossible, but that a particular combination of n and $\phi$ can produce predictions that are not observationally viable. If you increase the field value or use the e-fold-based estimate in the table, the numbers shift substantially.

As another check, look at the prefilled table below. For n = 2 and $N=60$, the helper function first estimates the field value from the e-fold relation and then computes $n_s$ and $r$. Those values are much closer to the standard textbook estimates for monomial slow-roll inflation. Comparing the manual form result with the table result is a good way to understand the difference between specifying $\phi$ directly and inferring it from $N$.

Interpretation of Results

When you read the output, start with $\epsilon$ and $\eta$. These are the basic indicators of whether the slow-roll approximation is self-consistent. Values much smaller than one generally support the approximation, while values approaching one suggest inflation is ending or the approximation is becoming unreliable. Next, inspect $n_s$. A value slightly below one corresponds to the observed red tilt of the scalar power spectrum. Finally, check $r$, which measures the relative strength of primordial tensor modes. Larger $r$ means a stronger gravitational-wave signal and usually a higher inflationary energy scale.

The page includes a small comparison against a commonly quoted observational upper bound on $r$. That message is useful as a quick screen, but it should not be treated as a full likelihood analysis. Real cosmological constraints depend on the exact dataset, confidence level, reheating assumptions, and whether the model is being compared at a fixed pivot scale. The calculator is best understood as a first-pass estimator rather than a substitute for a full parameter inference pipeline.

Reference Table and Trends

To illustrate the interplay between parameters, the table below lists slow-roll predictions for several choices of the power n and e-fold number $N$. The results show a familiar trend: larger n generally increases the tensor-to-scalar ratio, while larger $N$ tends to reduce $\epsilon$ and therefore lower $r$. These examples are generated automatically by the existing script and provide a quick benchmark for comparison with your own inputs.

The slow-roll formalism also helps estimate the duration of reheating, the period after inflation when the inflaton decays into standard particles. Different reheating histories effectively change the mapping between observable scales and the number of e-folds, thereby altering predictions for $n_s$ and $r$. While this calculator fixes $N$ as an input and uses it mainly in the helper function, more advanced analyses treat it as a derived quantity that depends on the thermalization temperature and the equation of state during reheating.

In addition to scalar and tensor spectra, slow-roll parameters influence higher-order statistics such as the bispectrum. In simple single-field models, non-Gaussianity is usually suppressed and often scales schematically like . That is one reason the slow-roll framework remains so central: the same small parameters that govern the background evolution also shape many observable signatures.

Limitations and Assumptions

This calculator is intentionally simple, and its limitations are important. First, it assumes a canonical single-field inflation model with a smooth monomial potential. It does not handle multifield dynamics, non-canonical kinetic terms, sharp features, turns in field space, or potentials with plateaus and inflection points where higher-order corrections may matter. Second, the displayed formulas are leading-order slow-roll expressions. They are excellent for intuition and rough estimates, but precision work may require next-order corrections and a more careful treatment of the horizon-crossing conditions.

Third, the page mixes two related but distinct ways of specifying the inflationary state. The form lets you enter $\phi$ directly, while the helper function used for the table estimates $\phi$ from $N$. That is not a bug in the script, but it does mean you should be clear about which quantity you are treating as fundamental in your own use. If you want a self-consistent monomial model at a chosen e-fold count, use the e-fold relation to guide your field choice. If you simply want to inspect the slow-roll parameters at a particular field value, the form output is the relevant result.

Finally, observational interpretation should be cautious. The page compares $r$ with a single threshold, but real cosmological constraints involve full datasets and model assumptions. A result that looks acceptable here is not automatically a viable inflation model, and a result that fails the quick comparison may still motivate theoretical study in a broader context. The calculator is best used for learning, screening, and building intuition before moving on to detailed numerical or statistical analysis.

By providing an interactive tool, this page bridges the gap between textbook formulas and hands-on exploration. Students can vary n, $\phi$, and $N$ to see how the resulting values of $n_s$ and $r$ change, building intuition about which models survive observational scrutiny. Researchers may also use it as a quick estimator when sketching out new ideas. Even in an era of precision cosmology, simple slow-roll relations remain one of the clearest ways to connect an inflaton potential to observable consequences.